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Analytic Solutions of Von Karman Plate

under Arbitrary Uniform PressurePart (II): Equations in Integral Form

Xiaoxu Zhong 3, Shijun Liao 1,2,3 ∗

1 State Key Laboratory of Ocean Engineering, Shanghai 200240, China2 Collaborative Innovative Center for Advanced Ship and Deep-Sea

Exploration, Shanghai 200240, China3 School of Naval Architecture, Ocean and Civil EngineeringShanghai Jiao Tong University, Shanghai 200240, China

Abstract In this paper, the homotopy analysis method (HAM) is successfullyapplied to solve the Von Karman’s plate equations in the integral form for acircular plate with the clamped boundary under an arbitrary uniform externalpressure. Two HAM-based approaches are proposed. One is for a given exter-nal load Q, the other for a given central deflection. Both of them are valid foran arbitrary uniform external pressure by means of choosing a proper valueof the so-called convergence-control parameters c1 and c2 in the frame of theHAM. Besides, it is found that iteration can greatly accelerate the conver-gence of solution series. In addition, we prove that the interpolation iterativemethod [1, 2] is a special case of the HAM-based 1st-order iteration approachfor a given external load Q when c1 = −θ and c2 = −1, where θ denotes theinterpolation parameter of the interpolation iterative method. Therefore, likeZheng and Zhou [3], one can similarly prove that the HAM-based approachesare valid for an arbitrary uniform external pressure, at least in some spe-cial cases such as c1 = −θ and c2 = −1. Furthermore, it is found that theHAM-based iteration approaches converge much faster than the interpolationiterative method [1, 2]. All of these illustrate the validity and potential of theHAM for the famous Von Karman’s plate equations, and show the superiorityof the HAM over perturbation methods.

Key Words circular plate, uniform external pressure, homotopy analysismethod

∗Corresponding author. Email address: sjliao@sjtu.edu.cn

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1. Introduction

In Part (I), we solved the Von Karman’s plate equations [4] in the differ-ential form for a circular plate under an arbitrary uniform external pressureby means of the homotopy analysis method (HAM) [5–11], an analyticalapproximation method for highly nonlinear problems. By means of the so-called convergence-control parameter c0, convergent analytic approximationsfor four kinds of boundaries were obtained, with large enough ratio of centraldeflection to thickness w(0)/h > 20. It is found that the convergence-controlparameter c0 plays an important role: it is the convergence-control param-eter c0 that can guarantee the convergence of solution series and iteration,and thus distinguishes the HAM from other analytic methods. Besides, itis found that the iteration technique can greatly increase the computationalefficiency. In addition, it was proved in Part (I) that the perturbation meth-ods for any a perturbation quantity (including Vincent’s [12] and Chien’s[13] perturbation methods) and the modified iteration method [14] are onlythe special cases of the HAM when c0 = −1.

The Von Karman’s plate equations [4] in the integral form describing thelarge deflection of a circular thin plate under a uniform external pressureread

ϕ(y) = −

∫

1

0

K(y, ε) ·1

ε2· S(ε)ϕ(ε)dε−

∫

1

0

K(y, ε) ·Qdε, (1)

S(y) =1

2

∫

1

0

G(y, ε) ·1

ε2· ϕ2(ε)dε, (2)

in which

K(y, ε) =

{

(λ− 1)yε+ y, y ≤ ε,(λ− 1)yε+ ε, y > ε,

(3)

G(y, ε) =

{

(µ− 1)yε+ y, y ≤ ε,(µ− 1)yε+ ε, y > ε,

(4)

with the definitions

y =r2

R2a

, W (y) =√

3(1− ν2)w(y)

h, ϕ(y) = y

dW (y)

dy, (5)

S(y) = 3(1− ν2)R2

aNr

Eh3y, Q =

3(1− ν2)√

3(1− ν2)R4a

4Eh4p, (6)

2

where r is the radial coordinate whose origin locates at the center of the plate,E, ν, Ra, h, w(y), Nr and p are Young’s modulus of elasticity, the Poisson’sratio, radius, thickness, deflection, the radial membrane force of the plateand the external uniform load, respectively, λ and µ are parameters relatedto the boundary conditions at y = 1, Q is a constant related to the uniformexternal load, respectively. The dimensionless central deflection

W (y) = −

∫

1

y

1

zϕ(z)dz (7)

can be derived from Eq.(5). Four kinds of boundaries are considered:(a) Clamped: λ = 0 and µ = 2/(1− ν);(b) Moveable clamped: λ = 0 and µ = 0;(c) Simple support: λ = 2/(1 + ν) and µ = 0;(d) Simple hinged support: λ = 2/(1 + ν) and µ = 2/(1− ν).Keller and Reiss [1] proposed the interpolation iterative method to solve

the Von Karman’s plate equations in the integral form by introducing aninterpolation parameter to the iteration procedure, and they successfullyobtained convergent solutions for loads as high as Q = 7000. Further, Zhengand Zhou [2, 3] proved that convergent solutions can be obtained by theinterpolation iterative method for an arbitrary uniform external pressure.Their excellent work is a milestone in this field.

Iterative procedures of the interpolation iterative method [1, 2] for theVon Karman’s plate equations in the integral form are:

ψn(y) =1

2

∫

1

0

G(y, ε) ·1

ε2· ϑ2n(ε)dε, (8)

ϑn+1(y) = (1− θ)ϑn(y)− θ

∫

1

0

K(y, ε)Qdε (9)

− θ

∫

1

0

K(y, ε) ·1

ε2· ϑn(ε)ψn(ε)dε.

with the definition of the initial guess

ϑ1(y) = −Qθ

2[(λ + 1)y − y2]. (10)

The homotopy analysis method (HAM) [5–11] was proposed by Liao [5]in 1992. Unlike perturbation technique, the HAM is independent of any

3

small/large physical parameters. Besides, unlike other analytic techniques,the HAM provides a simple way to guarantee the convergence of solutionseries by means of introducing the so-called “convergence-control parameter”.In addition, the HAM provides us great freedom to choose equation-typeand solution expression of the high-order linear equations. As a powerfultechnique to solve highly nonlinear equations, the HAM was successfullyemployed to solve various types of nonlinear problems over the past twodecades [15–25]. Especially, as shown in [26–30], the HAM can bring ussomething completely new/different: the steady-state resonant waves werefirst predicted by the HAM in theory and then confirmed experimentally ina lab [28].

In this paper, we propose two approaches in the frame of the HAM tosolve the Von Karman’s plate equations in the integral form. Besides, weproof that the interpolation iterative method [1, 2] is only a special case ofthe HAM.

2. HAM approach for given external load Q

2.1. Mathematical formulas

Following Zheng [2], we express ϕ(y) and S(y) as

ϕ(y) =

+∞∑

k=1

ak · yk, S(y) =

+∞∑

k=1

bk · yk, (11)

where ak and bk are constant coefficients to be determined. They provideus the so-called “solution expression” of ϕ(y) and S(y) in the frame of theHAM.

Let ϕ0(y) and S0(y) be initial guesses of ϕ(y) and S(y). Moreover, letc1 and c2 denote the non-zero auxiliary parameters, called the convergence-control parameters, and q ∈ [0, 1] the embedding parameter, respectively.Then we construct a family of differential equations in q ∈ [0, 1], namely thezeroth-order deformation equation:

(1− q)[Φ(y; q)− ϕ0(y)] = c1 q N1(y; q), (12)

(1− q)[Ξ(y; q)− S0(y)] = c2 q N2(y; q), (13)

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where

N1(y; q) = Φ(y; q) +

∫

1

0

K(y, ε) ·1

ε2· Φ(ε; q)Ξ(ε; q)dε

+

∫

1

0

K(y, ε) ·Qdε, (14)

N2(y; q) = Ξ(y; q)−1

2

∫

1

0

G(y, ε) ·1

ε2· Φ2(ε; q)dε, (15)

are two nonlinear operators.When q = 0, Eqs. (12) and (13) have the solution

Φ(y; 0) = ϕ0(y), Ξ(y; 0) = S0(y). (16)

When q = 1, they are equivalent to the original equations (1)-(2), provided

Φ(y; 1) = ϕ(y), Ξ(y; 1) = S(y). (17)

Therefore, as q increases from 0 to 1, Φ(y; q) varies continuously from theinitial guess ϕ0(y) to the solution ϕ(y), so does Ξ(y; q) from the initial guessS0(y) to the solution S(y). In topology, these kinds of continuous variationare called deformation. That is why Eqs. (12)-(13) constructing the homo-topies Φ(y; q) and Ξ(y; q) are called the zeroth−order deformation equations.

Expanding Φ(y; q) and Ξ(y; q; a) into Taylor series with respect to theembedding parameter q, we have the so-called homotopy-series:

Φ(y; q) = ϕ0(y) ++∞∑

k=1

ϕk(y)qk, (18)

Ξ(y; q) = S0(y) ++∞∑

k=1

Sk(y)qk, (19)

whereϕk(y) = Dk[Φ(y; q)], Sk(y) = Dk[Ξ(y; q)], (20)

in which

Dk[f ] =1

k!

∂kf

∂qk

∣

∣

∣

∣

q=0

(21)

is called the kth-order homotopy-derivative of f .

5

Note that there are two convergence-control parameters c1 and c2 in thehomotopy-series (18) and (19), respectively. Assume that c1 and c2 are prop-erly chosen so that the homotopy-series (18) and (19) converge at q = 1.Then according to Eq. (17), the so-called homotopy-series solutions read:

ϕ(y) = ϕ0(y) +

+∞∑

k=1

ϕk(y), (22)

S(y) = S0(y) +

+∞∑

k=1

Sk(y). (23)

The nth-order approximation of ϕ(y) and S(y) read

Φn(y) =

n∑

k=0

ϕk(y), (24)

Ξn(y) =n

∑

k=0

Sk(y). (25)

Substituting the power series (18) and (19) into the zeroth-order defor-mation equations (12) and (13), and then equating the like-power of theembedding parameter q, we have the so-called kth-order deformation equa-tions

ϕk(y)− χkϕk−1(y) = c1 δ1,k−1(y), (26)

Sk(y)− χkSk−1(y) = c2 δ2,k−1(y), (27)

where

δ1,k−1(y) = Dk−1[N1(y; q)]

= ϕk−1(y) +

∫

1

0

K(y, ε) ·1

ε2·

k−1∑

i=0

ϕi(ε)Sk−1−i(ε)dε

+ (1− χk)

∫

1

0

K(y, ε) ·Qdε, (28)

δ2,k−1(y) = Dk−1[N2(y; q)]

= Sk−1(y)−1

2

∫

1

0

G(y, ε) ·1

ε2·

k−1∑

i=0

ϕi(ε)ϕk−1−i(ε)dε, (29)

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with the definition

χm =

{

1 when m = 1,0 when m ≥ 2.

(30)

Note that the kth-order deformation equations (26)-(27) are linear.Note that we have great freedom to choose the initial guesses ϕ0(y) and

S0(y). According to [6], ϕ0(y) and S0(y) should obey the solution expression(11), thus, we choose the initial guesses:

ϕ0(y) =Qc02

[(λ+ 1)y − y2], (31)

S0(y) = 0, (32)

which satisfy all boundary conditions. Besides, for the sake of simplicity, set

c1 = c2 = c0. (33)

In order to measure the accuracy of approximations, we define the sum ofthe two discrete squared residuals

Err =1

K + 1

K∑

i=0

{

[

N1

(

i

K

)]2

+

[

N2

(

i

K

)]2}

, (34)

where N1 and N2 are two nonlinear operators defined by (14) and (15),respectively, and K = 100 is used in the whole of this paper. Obviously, thesmaller the Err, the more accurate the approximation.

2.2. Convergent results given by the HAM approach without iteration

Without loss of generality, the Von Karman’s plate equations with theclamped boundary is studied at first, and the Poisson’s ratio ν is taken to be0.3 in all cases considered in this paper.

First of all, the HAM-based approach (without iteration) is used to solvethe Von Karman’s plate equations for a given external load Q with theclamped boundary. Take the case of Q = 5 as an example. The optimalvalue of c0 is determined by the minimum of Err defined by (34), i.e. thesum of the squared residual errors of the two governing equations. As shownin Fig. 1, the sum of the squared residual errors arrives its minimum atc0 ≈ −0.35, which suggests that the optimal value of c0 is about −0.35.As shown in Table 1, the sum of the squared residual errors quickly de-creases to 1.7 × 10−7 by means of c0 ≈ −0.35 in the case of Q = 5, with

7

Table 1: The sum of the squared residual errors Err and the central deflection w(0)/hversus the order of approximation in the case of Q = 5 for a circular plate with the clampedboundary, given by the HAM approach (see § 2.1) without iteration using c0 = −0.35.

m, order of approx. Err w(0)/h10 3.3× 10−4 0.6420 6.5× 10−5 0.6230 1.1× 10−5 0.6240 1.6× 10−6 0.6250 1.7× 10−7 0.62

the convergent ratio of the maximum central deflection to plate thicknessw(0)/h = 0.62. Note that Vincent’s perturbation result [12] (using Q asthe perturbation quantity) for a circular plate with the clamped boundary isonly valid for w(0)/h < 0.52, corresponding to Q < 3.9, and thus fails in thecase of Q = 5. So, the convergence control parameter c0 indeed provides usa simple way to guarantee the convergence of solution series. This illustratesthat our HAM-based approach (mentioned in § 2.1) has advantages over theperturbation method [12].

For given values of Q, we can always find its optimal convergence con-trol parameter c0 in a similar way, which can be expressed by the empiricalformula:

c0 = −13

13 +Q2. (35)

Besides, the convergent homotopy-approximations of central deflection w(0)/hin case of different values of Q for a circular plate with the clamped boundaryare given in Table 2.

2.3. Convergence acceleration by iteration

According to Liao [6], the convergence of the homotopy-series solutionscan be greatly accelerated by means of iteration technique, which uses the

8

Table 2: The convergent results of the central deflection w(0)/h in the case of the differentvalues of Q for a circular plate with the clamped boundary, given by the HAM-basedapproach (see § 2.1) without iteration using the optimal convergence-control parameter c0given by (35).

Q c0 w(0)/h1 -0.93 0.152 -0.76 0.293 -0.59 0.414 -0.45 0.535 -0.34 0.62

c0

Err

-0.8 -0.6 -0.4 -0.2 010-2

10-1

100

101

102

103

104

Figure 1: The sum of the squared residual errors Err versus c0 in the case of the givenexternal load Q = 5 for a circular plate with the clamped boundary, given by the HAM-based approach (without iteration).

9

Mth-order homotopy-approximations

ϕ∗(y) ≈ ϕ0(y) +M∑

k=1

ϕk(y), (36)

S∗(y) ≈ S0(y) +

M∑

k=1

Sk(y), (37)

as the new initial guesses of ϕ0(y) and S0(y) for the next iteration. Thisprovides us the Mth-order iteration approach of the HAM.

Note that the length of the solution expressions increases exponentiallyin iteration. To avoid this, we truncate the right-hand sides of Eqs.(26) and(27) in the following way

δ1,k(y) ≈N∑

m=0

Ak,m · yk, δ2,k(y; a) ≈N∑

m=0

Bk,m · yk, (38)

where Ak,m and Bk,m are constant coefficients, and N is called the truncationorder, respectively.

Without loss of generality, let us consider the case of Q = 132.2, corre-sponding to w(0)/h = 3.0. As shown in Fig. 2, the higher the order M ofiteration, the less times of iteration is required for a given accuracy-level ofapproximation. Besides, the sum of the squared residual errors Err versusthe CPU time for different order of iteration (M) are given in Fig. 3. Notethat the higher the order of iteration, the faster the approximation con-verges. Thus, it is natural for us to choose the 5th-order iteration approach(i.e. M = 5) from the view-point of computational efficiency.

For a given Q, we use the truncation order N = 100. It is found that thecorresponding optimal convergence-control parameter c0 can be expressed bythe empirical formula:

c0 = −23

Q + 23. (39)

As shown in Table 3, convergent analytic solutions can be obtained even inthe case of Q = 1000, corresponding to w(0)/h = 6.1. Here, it should beemphasized that Vincent’s perturbation result [12] (using Q as the perturba-tion quantity) for a circular plate with the clamped boundary is only validfor w(0)/h < 0.52, corresponding to Q < 3.9 only. This illustrates again the

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Times of iteration

Err

0 10 20 30 4010-25

10-20

10-15

10-10

10-5

100

105

M = 1M = 2M = 3M = 4M = 5

Q = 132.5N = 100c0 = -0.15

Figure 2: The sum of the squared residual errors Err versus the times of iteration in thecase of Q = 132.2 for a circular plate with the clamped boundary, given by the HAM-based iteration approach (with different orders of iteration) using the convergence-controlparameter c0 = −0.15 and the truncation order N = 100. Square: 1st-order; Triangledown: 2nd-order; Circle: 3rd-order; Triangle up: 4th-order; Triangle left: 5th-order.

superiority of the HAM over perturbation methods. The convergent approx-imations of central deflection w(0)/h in case of different values of Q for acircular plate with the clamped boundary are listed in Table 4. In addition,several deflection curves under different loads Q are shown in Fig. 4.

2.4. Relations between the HAM and the interpolation iterative method

Here we prove that the interpolation iterative method [1, 2] is a specialcase of the HAM-based 1st-order iteration approach.

Let θ denote the interpolation parameter in the interpolation iterativemethod [1, 2]. Set c1 = −θ and c2 = −1, since we have the great freedomto choose their values in the frame of the HAM. According to (26)-(30) and(36)-(37), we have the 1st-order homotopy-approximations of ϕ(y) and S(y):

ϕ∗(y) = ϕ0(y) + ϕ1(y)

= ϕ0(y)− θδ1,1(y)

= (1− θ)ϕ0(y)− θ

∫

1

0

K(y, ε)Qdε

− θ

∫

1

0

K(y, ε) ·1

ε2· ϕ0(ε)S0(ε)dε, (40)

11

CPU time

Err

0 500 1000 1500 2000 2500 300010-25

10-20

10-15

10-10

10-5

100

105

M = 1M = 2M = 3M = 4M = 5

Q = 132.2N = 100c0 = -0.15

Figure 3: The sum of the squared residual errors Err versus the CPU time in the caseof Q = 132.2 for a circular plate with the clamped boundary, given by the HAM-basediteration approach (with different orders of iteration) using the convergence-control pa-rameter c0 = −0.15 and the truncation order N = 100. Square: 1st-order; Triangle down:2nd-order; Circle: 3rd-order; Triangle up: 4th-order; Triangle left: 5th-order.

r/Ra

w(0

)/h

0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

Q=100Q=200Q=500Q=1000

Figure 4: The deflection curves of a circular plate with the clamped boundary given bythe HAM-based 5th-order iteration approach in the case of Q = 100, 200, 500, 1000. Solidline: Q = 100; Dash-double-dotted line: Q = 200; Dash-dotted line: Q = 500; Dashedline: Q = 1000.

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Table 3: The sum of the squared residual errors Err and the homotopy-approximationsof w(0)/h at the mth iteration in the case of Q = 1000 for a circular plate with theclamped boundary, given by the HAM-based 5th-order iteration approach with the optimalconvergence-control parameter c0 = −0.02 and the truncation order N = 100.

m, times of iteration Err w(0)/h20 2.0× 10−1 6.140 1.4× 10−3 6.160 1.4× 10−5 6.180 1.4× 10−7 6.1100 1.5× 10−9 6.1

Table 4: The convergent homotopy-approximation of the central deflection w(0)/h incase of different values of Q for a circular plate with the clamped boundary, given by theHAM-based 5th-order iteration approach using the optimal convergence-control parameterc0 given in (39) and the truncation order N = 100.

Q c0 w(0)/h200 -0.10 3.5400 -0.05 4.5600 -0.04 5.2800 -0.03 5.71000 -0.02 6.1

S∗(y) = S0(y) + S1(y)

= S0(y)− δ2,1(y)

=1

2

∫

1

0

G(y, ε) ·1

ε2· ϕ2

0(ε)dε. (41)

Assume that ϕ0(y) and S0(y) are known. We use the following iterativeprocedure:

(A) Calculate S∗(y) according to Eq. (41);

(B) S0(y) is replaced by S∗(y) that is used as the new initial guess;

(C) Calculate ϕ∗(y) according to Eq. (40);

(D) ϕ0(y) is replaced by ϕ∗(y) that is used as the new initial guess.

13

At the nth times of iteration, write

Φn(y) = ϕ∗(y), Ξn−1(y) = S∗(y).

Then, the procedure of the HAM-based 1st-order iteration approach is ex-pressed by

Ξn−1(y) =1

2

∫

1

0

G(y, ε) ·1

ε2· Φ2

n−1(ε)dε, (42)

Φn(y) = (1− θ)Φn−1(y)− θ

∫

1

0

K(y, ε)Qdε

− θ

∫

1

0

K(y, ε) ·1

ε2· Φn−1(ε)Ξn−1(ε)dε. (43)

We choose the initial guess

Φ0(y) = −Qθ

2[(λ+ 1)y − y2]. (44)

Note that, the iterative procedures (8), (10) and the initial solution (10) ofthe interpolation iterative method [1, 2] are exactly the same as those of theHAM-based iteration approach (42)-(44). Thus, the interpolation iterativemethod [1, 2] is a special case of the HAM-based 1st-order iteration approachwhen c1 = −θ and c2 = −1. It should be emphasized that, as shown in Figs. 2and 3, the higher the order of the iteration, the faster the approximationsconverge. So, the interpolation iterative method [1, 2] should correspond tothe slowest one among the HAM-based Mth-order iteration approaches (upto M = 5). Finally, it should be emphasized that the interpolation iterativemethod [1, 2] is valid for an arbitrary uniform external pressure, as provedby Zheng and Zhou [3]. So, following Zheng and Zhou [3], one could provethat the HAM-based approach mentioned in § 2.1 is valid for an arbitraryuniform pressure at least in some special cases, such as c1 = −θ and c2 = −1.This reveals the important role of the convergence-control parameters c1 andc2 in the frame of the HAM.

3. HAM approach for given central deflection

According to Chien [13], it makes sense to introduce the central deflectioninto the Von Karman’s plate equations for a circular plate so as to enlargethe convergent region. Based on this knowledge, the HAM-based approachfor given central deflection is employed to solve the Von Karman’s plateequations in the integral form with the clamped boundary.

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3.1. Mathematical formulas

GivenW (0) = a,

we have due to Eq. (7) an additional equation for the corresponding unknownvalue of Q:

∫

1

0

1

εϕ(ε)dε = −a. (45)

Let ϕ0(y) and S0(y) denote initial guesses of ϕ(y) and S(y), which satisfythe restriction condition (45), c1 and c2 the non-zero auxiliary parameters,called the convergence-control parameters, q ∈ [0, 1] the embedding parame-ter, respectively. Then, we construct the so-called zeroth-order deformationequations

(1− q)[Φ(y; q)− ϕ0(y)] = c1 q N1(y; q), (46)

(1− q)[Ξ(y; q)− S0(y)] = c2 q N2(y; q), (47)

subject to the restricted condition∫

1

0

1

εΦ(ε; q)dε = −a. (48)

where

N1(y; q) = Φ(y; q) +

∫

1

0

K(y, ε) ·1

ε2· Φ(ε; q)Ξ(ε; q)dε

+

∫

1

0

K(y, ε) · Θ(q)dε, (49)

N2(y; q) = Ξ(y; q)−1

2

∫

1

0

G(y, ε) ·1

ε2· Φ2(ε; q)dε, (50)

are the two nonlinear operators, corresponding to Eqs. (1) and (2), respec-tively. Note that we introduce here a continuous variation Θ(q) from theinitial guess Q0 to Q, since Q is unknown for a given central deflection a.

When q = 0, it holds

Φ(y; 0) = ϕ0(y), Ξ(y; 0) = S0(y), (51)

since ϕ0(y) satisfies the restriction condition. When q = 1, they are equiva-lent to the original equations (1), (2) and (45), provided

Φ(y; 1) = ϕ(y), Ξ(y; 1) = S(y), Θ(1) = Q. (52)

15

Then, as q increases from 0 to 1, Φ(y; q) varies continuously from the initialguess ϕ0(y) to the solution ϕ(y), so do Ξ(y; q) from the initial guess S0(y) tothe solution S(y), Θ(q) from the initial guess Q0 to the unknown load Q forthe given central deflection a, respectively.

Expanding Φ(y; q), Ξ(y; q) and Θ(q) into Taylor series with respect to theembedding parameter q, we have the so-called homotopy-series:

Φ(y; q) = ϕ0(y) +

+∞∑

k=1

ϕk(y)qk, (53)

Ξ(y; q) = S0(y) ++∞∑

k=1

Sk(y)qk, (54)

Θ(q) = Q0 ++∞∑

k=1

Qkqk, (55)

in which

ϕk(y) = Dk[Φ(y; q)], Sk(y) = Dk[Ξ(y; q)], Qk = Dk[Θ(q)], (56)

with the definition Dk by (21). Assume that the homotopy-series (53)-(55)are convergent at q = 1. According to (52), we have the so-called homotopy-series solutions:

ϕ(y) = ϕ0(y) ++∞∑

k=1

ϕk(y), (57)

S(y) = S0(y) +

+∞∑

k=1

Sk(y), (58)

Q = Q0 +

+∞∑

k=1

Qk. (59)

Substituting the power series (53)-(55) into the zeroth-order deformationequations (46)-(48), and then equating the like-power of q, we have the so-called kth-order deformation equations

ϕk(y)− χkϕk−1(y) = c1 δ1,k−1(y), (60)

Sk(y)− χkSk−1(y) = c2 δ2,k−1(y), (61)

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subject to the restriction condition

∫

1

0

1

εϕk(ε)dε = 0, (62)

where χk is defined by (30), and

δ1,k−1(y) = Dk−1[N1(y; q)]

= ϕk−1(y) +

k−1∑

i=0

∫

1

0

K(y, ε) ·1

ε2· ϕi(ε)Sk−1−i(ε)dε

+

∫

1

0

K(y, ε) ·Qk−1dε, (63)

δ2,k−1(y) = Dk−1[N2(y; q)]

= Sk−1(y)−

k−1∑

i=0

1

2

∫

1

0

G(y, ε) ·1

ε2· ϕi(ε)ϕk−1−i(ε)dε, (64)

in which Qk−1 is determined by the restriction condition (62). Consequently,ϕk(y) and Sk(y) of Eqs. (60) and (61) are obtained. Then, we have thenth-order approximation:

ϕ(y) =

n∑

k=0

ϕk(y), (65)

S(y) =

n∑

k=0

Sk(y), (66)

Q =n

∑

k=0

Qk. (67)

According to [6], the initial guesses ϕ0(y) and S0(y) should obey the so-lution expression (11) and satisfy the boundary conditions. Thus, we choose

ϕ0(y) =−2a

2λ+ 1[(λ+ 1)y − y2], (68)

S0(y) = 0, (69)

as the initial guesses of ϕ(y) and S(y), respectively.

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c0

Err

-1 -0.8 -0.6 -0.4 -0.2 0100

101

102

103

104

105

106

Figure 5: The sum of the squared residual errors Err versus c0 in the case of a = 5 for acircular plate with the clamped boundary, gained by the HAM-based approach (withoutiteration).

3.2. Results given by the HAM-approach without iteration

First, the HAM-based approach (without iteration) for given central de-flection is used to solve the Von Karman’s plate equations in the integralform with the clamped boundary. Without loss of generality, let us considerthe same case of a = 5, equivalent to the central deflection w(0)/h = 3. Asshown in Fig. 5, the sum of the squared residual errors Err arrives its min-imum at c0 ≈ −0.25. As shown in Table 5, the sum of the squared residualerrors quickly decreases to 3.6 × 10−7 by means of c0 ≈ −0.25 in the caseof a = 5. Note that, the Chien’s perturbation method [13] (using a as theperturbation quantity) is only valid for w(0)/h < 2.44, equivalent to a < 4,for a plate with the clamped boundary. This again illustrates the superiorityof the HAM-based approach over the perturbation method.

Similarly, for given value of a, we can always find its optimal value of c0,which can be expressed by the empirical formula:

c0 = −11

11 + a2(a ≤ 5). (70)

In addition, the convergent results of the external load Q in case of differentvalues of a for a circular plate with the clamped boundary are given in

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Table 5: The sum of the squared residual errors Err and the corresponding load Q versusthe order of approximation in the case of a = 5 for a circular plate with the clampedboundary, given by the HAM without iteration using c0 = −0.25.

Order of approx. Err Q20 3.4× 10−2 132.340 2.2× 10−3 132.560 8.8× 10−5 132.380 9.7× 10−7 132.2100 3.6× 10−7 132.2

Table 6: The convergent results of the external load Q in case of different values of a fora circular plate with the clamped boundary, given by the HAM-based approach withoutiteration using the optimal convergence-control parameter c0 given by (70).

a c0 Q1 -0.92 4.82 -0.73 14.63 -0.55 35.24 -0.41 72.45 -0.31 132.2

Table 6.

3.3. Convergence acceleration by means of iteration

As shown in § 2.3, iteration can greatly accelerate the convergence of thehomotopy-series solutions. Therefore, we used the HAM-based iteration ap-proach for given central deflection to solve the Von Karman’s plate equationswith the clamped boundary in this subsection. The iteration order M andthe truncation order N are defined in a similar way to Eqs. (36) and (38).

The iteration order M = 5 and the truncation order N = 100 are usedin all cases described below. Without loss of generality, let us first considerthe same case of a = 5. Note that the sum of the squared residual errorsquickly decreases to 1.4 × 10−28 in only 10 iteration times, as shown in Ta-ble 7. In addition, as shown in Figs. 6 and 7, the convergence is much faster

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Table 7: The sum of the squared residual Err and the load Q versus the iteration times inthe case of a = 5 for a circular plate with the clamped boundary, given by the HAM-based5th-order iteration approach using c0 = −0.5 and the truncation order N = 100.

m, times of iteration. Err Q2 2.0× 10−3 132.04 8.4× 10−9 132.26 1.7× 10−16 132.28 5.3× 10−22 132.210 1.4× 10−28 132.2

by introducing W (0) into the Von Karman’s plate equations. Therefore,using the central deflection indeed makes sense. It should be emphasizedthat the HAM-based iteration approaches converges much faster than theinterpolation iterative method [1, 2], as shown in Figs. 6 and 7.

Similarly, for any a given value of a, we can always find the optimalvalue of the convergence-control parameter c0, which can be expressed bythe empirical formula:

c0 = −25

25 + a2. (71)

As shown in Table 8, convergent solutions can be obtained even in the caseof a = 30, equivalent to w(0)/h = 18.2. As shown in Fig. 8, as a increases,the interpolation parameter θ of the interpolation iterative method [1, 2]approaches to 0 much faster than the optimal convergence-control parameterc0 mentioned-above. This explains why the HAM-based iteration approachesconverge much faster than the interpolation iterative method [1, 2].

4. Concluding remarks

In this paper, the homotopy analysis method (HAM) is successfully ap-plied to solve the Von Karman’s plate equations in the integral form for acircular plate with the clamped boundary under an arbitrary uniform exter-nal pressure. Two HAM-based approached are proposed. One is for givenload Q, the other for given central deflection. Both of them are valid for ar-bitrary uniform external pressure, by mens of choosing a proper value of theso-called convergence-control parameters c1 and c2 in the frame of the HAM.

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CPU time

Err

0 500 1000 1500 2000 2500 300010-30

10-25

10-20

10-15

10-10

10-5

100

Figure 6: The sum of the squared residual errors Err versus the CPU time in the caseof a = 5 for a circular plate with the clamped boundary, gained by the interpolationiterative method [1, 2], and the HAM-based approach for given external load Q and centraldeflection a, respectively. Dash-dotted line: results given by the interpolation iterativemethod [1, 2] using the interpolation parameter θ = 0.1; Dashed line: results given bythe HAM-based 5th-order iteration approach for given external load Q using the optimalconvergence-control parameter c0 = −0.15 and the truncation order N = 100; Solid line:results given by the HAM-based 5th-order iteration approach for given central deflectiona using the optimal convergence-control parameter c0 = −0.5 and the truncation orderN = 100.

Besides, it is found that iteration can greatly accelerate the convergenceof solution series. In addition, it is proved that the interpolation iterativemethod [1, 2] is a special case of the HAM-based 1st-order iteration approachfor given external load Q when c1 = −θ and c2 = −1, where θ denotes the in-terpolation parameter of the interpolation iterative method†. Therefore, likeZheng and Zhou [3], one can similarly prove that the HAM-based approachesfor the Von Karman’s plate equations in the integral form are valid for anarbitrary uniform external pressure, at least in some special cases such asc1 = −θ and c2 = −1. Finally, it should be emphasized that the HAM-based

†Note that, it was proved in Part (I) that the perturbation methods for any a pertur-bation quantity (including Vincent’s [12] and Chien’s [13] perturbation methods) and themodified iteration method [14] are only the special cases of the HAM-based approacheswhen c0 = −1, for the Von Karman’s plate equations in the differential form for a circularplate under a uniform external pressure.

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Times of iteration

Err

0 10 20 30 4010-30

10-25

10-20

10-15

10-10

10-5

100

Figure 7: The sum of the squared residual errors Err versus iteration times in the caseof a = 5 for a circular plate with the clamped boundary, gained by the interpolationiterative method [1, 2], and the HAM-based approach for given external load Q and centraldeflection a, respectively. Dash-dotted line: results given by the interpolation iterativemethod [1, 2] using the interpolation parameter θ = 0.1; Dashed line: results given bythe HAM-based 5th-order iteration approach for given external load Q using the optimalconvergence-control parameter c0 = −0.15 and the truncation order N = 100; Solid line:results given by the HAM-based 5th-order iteration approach for given central deflectiona using the optimal convergence-control parameter c0 = −0.5 and the truncation orderN = 100.

Table 8: The convergent homotopy-approximation of Q in case of different values of afor a circular plate with the clamped boundary, given by the HAM-based 5th-order iter-ation approach with the optimal convergence-control parameter c0 given by (71) and thetruncation order N = 100.

a c0 Q5 -0.50 132.210 -0.20 957.715 -0.10 3152.120 -0.06 7386.925 -0.04 14334.130 -0.03 24665.7

22

a

Con

verg

ence

-con

trol

par

amet

er

2 4 6 8 10

-1

-0.8

-0.6

-0.4

-0.2

0

c0

-

Figure 8: The convergence-control parameter c0 and the interpolation parameter θ versusthe central deflection a for a circular plate with the clamped boundary, gained by theinterpolation iterative method [2] for given external load Q and the HAM-based 5th-orderiteration approach for given central deflection a using the truncation order N = 100.Dashed line: −θ; Solid line: c0.

iteration approaches converge much faster than the interpolation iterativemethod [1, 2], as shown in Figs. 6 and 7. All of these illustrate the validityand potential of the HAM for the famous Von Karman’s plate equations,and show the superiority of the HAM over perturbation methods. Withoutdoubt, the HAM can be applied to solve some challenging nonlinear problemsin solid mechanics.

Acknowledgment

This work is partly supported by National Natural Science Foundation ofChina (Approval No. 11272209 and 11432009) and State Key Laboratory ofOcean Engineering (Approval No. GKZD010063).

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